Solution
= Solution
If $P\,dx+Q\,dy=dF$ with continuous second <partial derivatives> of $F$, then $P=F_x$, $Q=F_y$. The <symmetry of second derivatives> requires
$$
\boxed{P_y=Q_x.}
$$
This is a necessary condition for an <exact differential>. With continuously <differentiable> <coefficients> it is also locally sufficient, and is globally sufficient on a simply connected open domain; a domain with holes can require additional global information.